The gradient structure of the mean curvature flow

2015 ◽  
Vol 8 (3) ◽  
Author(s):  
Martijn M. Zaal
Keyword(s):  
Metric Space ◽  
Mean Curvature ◽  
Metric Spaces ◽  
Mean Curvature Flow ◽  
Gradient Flow ◽  
Curvature Flow ◽  
Gradient Structure ◽  
Length Space ◽  
Curve Length ◽  
The Mean

AbstractThe concept of curve of maximal slope, a generalized notion of gradient flow, is extended to the setting of length spaces, which includes all metric spaces. This extended definition is used to show that curves of maximal slope in a metric space do not depend on the full metric, but only on the concept of curve length generated by it. Subsequently, it is shown that a length space can be constructed to describe

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

scholarly journals Finite topology self-translating surfaces for the mean curvature flow in R3

2017 ◽  
Vol 320 ◽  
pp. 674-729 ◽  
Author(s):  
Juan Dávila ◽  
Manuel del Pino ◽  
Xuan Hien Nguyen
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Finite Topology ◽  
The Mean

The space of compact self-shrinking solutions to the Lagrangian mean curvature flow in ℂ 2 {\mathbb{C}^{2}}

2018 ◽  
Vol 2018 (743) ◽  
pp. 229-244 ◽  
Author(s):  
Jingyi Chen ◽  
John Man Shun Ma
Keyword(s):  
Upper Bound ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Riemannian Metric ◽  
Branch Points ◽  
Rigidity Result ◽  
The Mean ◽  
Uniform Area

Abstract Let F_{n} : (Σ, h_{n} ) \to \mathbb{C}^{2} be a sequence of conformally immersed Lagrangian self-shrinkers with a uniform area upper bound to the mean curvature flow, and suppose that the sequence of metrics \{ h_{n} \} converges smoothly to a Riemannian metric h. We show that a subsequence of \{ F_{n} \} converges smoothly to a branched conformally immersed Lagrangian self-shrinker F_{\infty} : (Σ, h) \to \mathbb{C}^{2} . When the area bound is less than 16π, the limit {F_{\infty}} is an embedded torus. When the genus of Σ is one, we can drop the assumption on convergence h_{n} \to h. When the genus of Σ is zero, we show that there is no branched immersion of Σ as a Lagrangian self-shrinker, generalizing the rigidity result of [21] in dimension two by allowing branch points.

Rotational λ – hypersurfaces in Euclidean spaces

2021 ◽  
Vol 30 (1) ◽  
pp. 29-40
Author(s):  
KADRI ARSLAN ◽  
ALIM SUTVEREN ◽  
BETUL BULCA
Keyword(s):  
Present Article ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Special Solution ◽  
Euclidean Spaces ◽  
Self Similar ◽  
The Mean ◽  
Similar Flows

Self-similar flows arise as special solution of the mean curvature flow that preserves the shape of the evolving submanifold. In addition, \lambda -hypersurfaces are the generalization of self-similar hypersurfaces. In the present article we consider \lambda -hypersurfaces in Euclidean spaces which are the generalization of self-shrinkers. We obtained some results related with rotational hypersurfaces in Euclidean 4-space \mathbb{R}^{4} to become self-shrinkers. Furthermore, we classify the general rotational \lambda -hypersurfaces with constant mean curvature. As an application, we give some examples of self-shrinkers and rotational \lambda -hypersurfaces in \mathbb{R}^{4}.

scholarly journals The mean curvature flow by parallel hypersurfaces

2018 ◽  
Vol 146 (11) ◽  
pp. 4867-4878 ◽  
Author(s):  
Hiuri Fellipe Santos dos Reis ◽  
Keti Tenenblat
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
The Mean ◽  
Parallel Hypersurfaces
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